Scalar Differential Invariants of Symplectic Monge-Amp\`ere Equations

TL;DR

This paper explicitly computes all second order scalar differential invariants of symplectic Monge-Ampère equations, revealing a higher number of invariants than in general cases and providing tools for their classification and linearization.

Contribution

It introduces a complete set of second order invariants for symplectic Monge-Ampère equations and constructs higher order invariants using invariant forms and vector fields.

Findings

Number of second order invariants is 7 for symplectic cases.

Constructed invariant differential forms and vector fields for higher order invariants.

Provided a linearization criterion for certain quasilinear equations.

Abstract

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Amp\`ere equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Amp\`ere equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allows us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Amp\`ere equations. As an example we study quasilinear equations of a suitable kind and in particular find a simple linearization criterion.